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A320750 Array read by antidiagonals: T(n,k) is the number of color patterns (set partitions) in an unoriented row of length n using k or fewer colors (subsets). 7
1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 4, 6, 1, 1, 2, 4, 10, 10, 1, 1, 2, 4, 11, 25, 20, 1, 1, 2, 4, 11, 31, 70, 36, 1, 1, 2, 4, 11, 32, 107, 196, 72, 1, 1, 2, 4, 11, 32, 116, 379, 574, 136, 1, 1, 2, 4, 11, 32, 117, 455, 1451, 1681, 272, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

Two color patterns are equivalent if the colors are permuted.

In an unoriented row, chiral pairs are counted as one.

T(n,k) = Pi_k(P_n) which is the number of non-equivalent partitions of the path on n vertices, with at most k parts. Two partitions P1 and P2 of a graph G are said to be equivalent if there is a nontrivial automorphism of G which maps P1 onto P2. - Mohammad Hadi Shekarriz, Aug 21 2019

REFERENCES

B. Ahmadi, F. Alinaghipour and M. H. Shekarriz, Number of Distinguishing Colorings and Partitions, Preprint

LINKS

Table of n, a(n) for n=1..66.

FORMULA

T(n,k) = Sum_{j=1..k} (S2(n,j) + Ach(n,j))/2, where S2 is the Stirling subset number A008277 and Ach(n,k) = [n>=0 & n<2 & n==k] + [n>1]*(k*Ach(n-2,k) + Ach(n-2,k-1) + Ach(n-2,k-2)).

T(n,k) = (A278984(k,n) + A305749(n,k)) / 2 = A278984(k,n) - A320751(n,k) = A320751(n,k) + A305749(n,k).

T(n,k) = Sum_{j=1..k} A284949(n,j).

EXAMPLE

Array begins with T(1,1):

1   1     1     1      1      1      1      1      1      1      1 ...

1   2     2     2      2      2      2      2      2      2      2 ...

1   3     4     4      4      4      4      4      4      4      4 ...

1   6    10    11     11     11     11     11     11     11     11 ...

1  10    25    31     32     32     32     32     32     32     32 ...

1  20    70   107    116    117    117    117    117    117    117 ...

1  36   196   379    455    467    468    468    468    468    468 ...

1  72   574  1451   1993   2135   2151   2152   2152   2152   2152 ...

1 136  1681  5611   9134  10480  10722  10742  10743  10743  10743 ...

1 272  5002 22187  43580  55091  58071  58461  58486  58487  58487 ...

1 528 14884 87979 211659 301633 333774 339764 340359 340389 340390 ...

For T(4,3)=10, the patterns are AAAA, AABB, ABAB, ABBA, ABBC, ABCA, AAAB, AABA, AABC, ABAC, the last four being chiral with partners ABBB, ABAA, ABCC, and ABCB.

MATHEMATICA

Ach[n_, k_] := Ach[n, k] = If[n<2, Boole[n==k && n>=0], k Ach[n-2, k] + Ach[n-2, k-1] + Ach[n-2, k-2]] (* A304972 *)

Table[Sum[StirlingS2[n, j] + Ach[n, j], {j, k-n+1}]/2, {k, 15}, {n, k}] // Flatten

CROSSREFS

Columns 1-6 are A000012, A005418, A001998(n-1), A056323, A056324, A056325.

As k increases, columns converge to A103293(n+1).

Cf. transpose of A278984 (oriented), A320751 (chiral), A305749 (achiral).

Partial column sums of A284949.

Sequence in context: A152977 A259799 A208447 * A117935 A224698 A179749

Adjacent sequences:  A320747 A320748 A320749 * A320751 A320752 A320753

KEYWORD

nonn,tabl,easy

AUTHOR

Robert A. Russell, Oct 27 2018

STATUS

approved

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Last modified September 15 08:37 EDT 2019. Contains 327062 sequences. (Running on oeis4.)